Quantum Topology


Full-Text PDF (1542 KB) | Metadata | Table of Contents | QT summary
Volume 4, Issue 1, 2013, pp. 1–75
DOI: 10.4171/QT/34

Published online: 2012-11-24

A diagrammatic categorification of the q-Schur algebra

Marco Mackaay[1], Marko Stošić[2] and Pedro Vaz[3]

(1) Universidade do Algarve, Faro, Portugal
(2) Instituto Superior Técnico, Lisboa, Portugal
(3) Instituto Superior Técnico, Lisboa, Portugal

In this paper we categorify the q-Schur algebra $\mathbf{S}_q(n,d)$ as a quotient of Khovanov and Lauda’s diagrammatic 2-category $\mathcal{U}(\mathfrak{sl}_n)$ [16]. We also show that our 2-category contains Soergel’s [33] monoidal category of bimodules of type $A$, which categorifies the Hecke algebra $H_q(d)$, as a full sub-2-category if $d\leq n$. For the latter result we use Elias and Khovanov’s diagrammatic presentation of Soergel’s monoidal category of type $A$; see [8].

Keywords: Categorification, quantum groups, quantum $\mathfrak{gl}_n$, q-Schur algebra, Soergel category

Mackaay Marco, Stošić Marko, Vaz Pedro: A diagrammatic categorification of the q-Schur algebra. Quantum Topol. 4 (2013), 1-75. doi: 10.4171/QT/34